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Properties

Label 2-432-16.13-c1-0-5
Degree $2$
Conductor $432$
Sign $-0.923 - 0.382i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.00·4-s + (0.707 − 0.707i)5-s + 3i·7-s − 2.82i·8-s + (1.00 + 1.00i)10-s + (−3.53 + 3.53i)11-s + (1 + i)13-s − 4.24·14-s + 4.00·16-s − 4.24·17-s + (4 + 4i)19-s + (−1.41 + 1.41i)20-s + (−5.00 − 5.00i)22-s + 2.82i·23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + (0.316 − 0.316i)5-s + 1.13i·7-s − 1.00i·8-s + (0.316 + 0.316i)10-s + (−1.06 + 1.06i)11-s + (0.277 + 0.277i)13-s − 1.13·14-s + 1.00·16-s − 1.02·17-s + (0.917 + 0.917i)19-s + (−0.316 + 0.316i)20-s + (−1.06 − 1.06i)22-s + 0.589i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197338 + 0.992088i\)
\(L(\frac12)\) \(\approx\) \(0.197338 + 0.992088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
good5 \( 1 + (-0.707 + 0.707i)T - 5iT^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + (3.53 - 3.53i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + (-4 - 4i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (2.82 + 2.82i)T + 29iT^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T - 53iT^{2} \)
59 \( 1 + (-7.07 + 7.07i)T - 59iT^{2} \)
61 \( 1 + (-10 - 10i)T + 61iT^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 - 15.5iT - 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-7.77 - 7.77i)T + 83iT^{2} \)
89 \( 1 + 1.41iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.72239550204293097276932760191, −10.37288419473879299098878433959, −9.391505187784055293581236183049, −8.885596334948817280599607665446, −7.77621427101332911067581243297, −6.99280715791927028667814909058, −5.56852503776266656096625137062, −5.34424065956781770844509271726, −3.90117756379454380132469208167, −2.10015245681796227737590973184, 0.64392064367176100058651411442, 2.48700925966976332164969174878, 3.53168370886658310772605114420, 4.70251855247389179231846771630, 5.79636093441933437442307283107, 7.15257645919197829701276295688, 8.189647505455766750007102105390, 9.131417365339682549981965809734, 10.14617921843159344065175093574, 11.00034574808108996135335244119

Graph of the $Z$-function along the critical line